GLOBAL EDITION Using and Understanding Mathematics A Quantitative Reasoning Approach SIXTH EDITION Jeffrey Bennett William Briggs
Why Should you Care About Quantitative reasoning? Quantitative reasoning is the ability to interpret and reason with information that involves numbers or mathematical ideas. It is a crucial aspect of literacy, and it is essential in making important decisions and understanding contemporary issues. The topics covered in this text will help you work with quantitative information and make critical decisions. For example: You should possess strong skills in critical and logical thinking so that you can make wise personal decisions, navigate the media, and be an informed citizen. For example, do you know why you d end up behind if you accepted a temporary 10% pay cut now and then received a 10% pay raise later? This particular question is covered in Unit 3A, but throughout the book you ll learn how to evaluate quantitative questions on topics ranging from personal decisions to major global issues. You should have a strong number sense and be proicient at estimation so that you can put numbers from the news into a context that makes them understandable. For example, do you know how to make sense of the more than $17 trillion federal debt? Unit 3B discusses how you can put such huge numbers in perspective, and Unit 4F discusses how the federal debt grew so large. You should possess the mathematical tools needed to make basic inancial decisions. For example, do you enjoy a latte every morning before class? Sometimes two? Unit 4A explores how such a seemingly harmless habit can drain more than $2400 from your wallet every year. You should be able to read news reports of statistical studies in a way that will allow you to evaluate them critically and decide whether and how they should afect your personal beliefs. For example, how should you decide whether a new opinion poll accurately relects the views of Americans? Chapter 5 covers the basic concepts that lie behind the statistical studies and graphics you ll see in the news, and discusses how you can decide for yourself whether you should believe a statistical study. You should be familiar with basic ideas of probability and risk and be aware of how they afect your life. For example, would you pay $20,000 for a product that, over 20 years, will kill nearly as many people as live in San Francisco? In Unit 7D, you ll see that the answer is very likely yes just one of many surprises that you ll encounter as you study probability in Chapter 7. You should understand how mathematics helps us study important social issues, such as global warming, the growth of populations, the depletion of resources, apportionment of Congressional representatives, and methods of voting. For example, Unit 12D discusses the nature of redistricting and how gerrymandering has made congressional elections less competitive than they might otherwise be. In sum, this text will focus on understanding and interpreting mathematical topics to help you develop the quantitative reasoning skills you will need for college, career, and life.
230 chapter 4 Managing Money using TeCHNoLoGy The Compound interest Formula Standard Calculators You can do compound interest calculations on any calculator that has a key for raising numbers to powers ( y x or ). The only trick is making sure you follow the standard order of operations: 1. Parentheses: Do terms in parentheses first. 2. Exponents: Do powers and roots next. 3. Multiplication and Division: Work from left to right. 4. Addition and Subtraction: Work from left to right. Let s apply this order of operations to the compound interest problem from Example 2, in which we have P = $100, APR = 0.1, and Y = 5 years. General Procedure our example Calculator Steps output A = P * 11 + APR2 ('')'+* Y 1. parentheses (''')''+* 2. exponent ('''')'''+* 3. multiply A = 100 * 11 + 0.12 5 ('')'+* 1. parentheses (''')''+* 2. exponent ('''')'''+* 3. multiply Step 1 1 + 0.1 = Step 2 5 = Step 3 * 100 = 1.1 1.61051 161.051 note: Do not round answers in intermediate steps; only the final answer should be rounded to the nearest cent. excel Use the built-in function FV (for future value) for compound interest calculations in Excel. The screen shot to the right shows the use of this function for our sample calculation. The table at the bottom explains the inputs that go in the parentheses of the FV function. Note: You could get the final result by typing values directly into the FV function, but as shown in the screen shot, it is better to show your work. Here we put variable names in Column A and values in Column B, using the FV function in cell B5. Besides making your work clearer, this approach makes it easy to do what if scenarios, such as changing the interest rate or number of years. (''')''+* You can remember the order of operations with the mnemonic Please Excuse My Dear Aunt Sally. input Description our example rate The interest rate for each compounding period Because we are using interest compounded once a year, the interest rate is the annual rate, APR = 0.1. nper The total number of compounding periods For interest compounded once a year, the total number of compounding periods is the number of years, Y = 5. pmt pv type The amount of any payment made each month The present value, equivalent to the starting principal P An optional input related to whether monthly payments are made at the beginning 1type = 02 or end 1type = 12 of a month No payment is being made monthly in our example, so we enter 0. We use the starting principal, P = 100. Type does not apply in this case because there is no monthly payment, so we do not include it.
4b The Power of Compounding 231 compound interest as exponential growth The New College case demonstrates the remarkable way in which money can grow with compound interest. Figure 4.2 shows how the value of the New College debt rises during the first 100 years, assuming a starting value of $224 and an interest rate of 4% per year. Note that while the value rises slowly at first, it rapidly accelerates, so in later years the value grows by much more each year than it did during earlier years. $15,000 Accumulated value $10,000 $5000 This rapid growth is a hallmark of what we generally call exponential growth. You can see how exponential growth gets its name by looking again at the general compound interest formula: A = P * 11 + APR2 Y Because the starting principal P and the interest rate APR have fixed values for any particular compound interest calculation, the growth of the accumulated value A depends only on Y (the number of times interest has been paid), which appears in the exponent of the calculation. Exponential growth is one of the most important topics in mathematics, with applications that include population growth, resource depletion, and radioactivity. We will study exponential growth in much more detail in Chapter 8. In this chapter, we focus only on its applications in finance. example 3 new college debt at 2% If the interest rate is 2%, calculate the amount due to New College using a. simple interest b. compound interest Solution a. The following steps show the simple interest rate calculation for a starting principal P = $224 and an annual interest rate of 2%: 1. The simple interest due each year is 2% of the starting principal: 2% * $224 = 0.02 * $224 = $4.48 2. Over 535 years, the total interest due is: 535 * $4.48 = $2396.80 3. The total due after 535 years is the starting principal plus the interest: $224 0 20 40 60 80 100 FiGure 4.2 The value of the debt in the New College case during the first 100 years, at an interest rate of 4% per year. Note that the value rises much more rapidly in later years than in earlier years a hallmark of exponential growth. Years $224 + $2396.80 = $2620.80 With simple interest, the payoff amount after 535 years is $2620.80.
232 chapter 4 Managing Money b. To find the amount due with compound interest, we set the annual interest rate to APR = 2% = 0.02 and the number of years to Y = 535. Then we use the formula for compound interest paid once a year: A = P * 11 + APR2 Y = $224 * 11 + 0.022 535 = $224 * 11.022 535 $224 * 39,911 $8.94 * 10 6 The amount due with compound interest is about $8.94 million far higher than the amount due with simple interest. Now try exercises 57 58. effects of interest rate changes Notice the remarkable effects of small changes in the compound interest rate. In Example 3, we found that a 2% compound interest rate leads to a payoff amount of $8.94 million after 535 years. Earlier, we found that a 4% interest rate for the same 535 years leads to a payoff amount of $290 billion which is more than 30,000 times as large as $8.94 million. Figure 4.3 contrasts the values of the New College debt during the first 100 years at interest rates of 2% and 4%. Note that the rate change doesn t make much difference for the first few years, but over time the higher rate becomes far more valuable. $15,000 Accumulated value $10,000 $5000 APR 4% APR 2% $224 0 20 40 60 80 100 Years FiGure 4.3 This figure contrasts the debt in the New College case during the first 100 years at interest rates of 2% and 4%. Time Out to Think Suppose the interest rate for the New College debt were 3%. Without calculating, do you think the value after 535 years would be halfway between the values at 2% and 4% or closer to one or the other of these values? Now, check your guess by calculating the value at 3%. What happens at an interest rate of 6%? Briefly discuss why small changes in the interest rate can lead to large changes in the accumulated value. example 4 Mattress investments Your grandfather put $100 under his mattress 50 years ago. If he had instead invested it in a bank account paying 3.5% interest compounded yearly (roughly the average U.S. rate of inflation during that period), how much would it be worth now?
4b The Power of Compounding 233 Solution The starting principal is P = $100. The annual percentage rate is APR = 3.5% = 0.035. The number of years is Y = 50. So the accumulated balance is A = P * 11 + APR2 Y = $100 * 11 + 0.0352 50 = $100 * 11.0352 50 = $558.49 Invested at a rate of 3.5%, the $100 would be worth over $550 today. Unfortunately, the $100 was put under a mattress, so it still has a face value of only $100. Now try exercises 59 62. compound interest Paid More than once a year Suppose you could put $1000 into an investment that pays compound interest at an annual percentage rate of APR = 8%. If the interest is paid all at once at the end of a year, you ll receive interest of TABLe 4.3 8% * $1000 = 0.08 * $1000 = $80 Therefore, your year-end balance will be $1000 + $80 = $1080. Now, assume instead that the investment pays interest quarterly, or four times a year (once every 3 months). The quarterly interest rate is one-fourth of the annual interest rate: quarterly interest rate = APR 4 = 8% 4 = 2% = 0.02 Table 4.3 shows how quarterly compounding affects the $1000 starting principal during the first year. Quarterly interest Payments (P = $1000, apr = 8%) After N Quarters interest Paid New Balance 1st quarter (3 months) 2% * $1000 = $20 $1000 + $20 = $1020 2nd quarter (6 months) 2% * $1020 = $20.40 $1020 + $20.40 = $1040.40 3rd quarter (9 months) 2% * $1040.40 = $20.81 $1040.40 + $20.81 = $1061.21 4th quarter (1 full year) 2% * $1061.21 = $21.22 $1061.21 + $21.22 = $1082.43 Note that the year-end balance with quarterly compounding 1$1082.432 is greater than the year-end balance with interest paid all at once 1$10802. That is, when interest is compounded more than once a year, the balance increases by more than the APR in 1 year. We can find the same results with the compound interest formula. Remember that the basic form of the compound interest formula is A = P * 11 + interest rate2 number of compoundings where A is the accumulated balance and P is the starting principal. In our current case, the starting principal is P = $1000, the quarterly payments have an interest rate of APR>4 = 0.02, and in one year the interest is paid four times. Therefore, the accumulated balance at the end of one year is number of A = P * 11 + interest rate2 compoundings = $1000 * 11 + 0.022 4 = $1082.43 We see that if interest is paid quarterly, the interest rate at each payment is APR>4. Generalizing, if interest is paid n times per year, the interest rate at each payment is APR>n. The total number of times that interest is paid after Y years is ny. We therefore find the following formula for interest paid more than once each year.
234 chapter 4 Managing Money Compound interest Formula for interest Paid N Times Per year A = P a1 + APR n 1nY2 b where A = accumulated balance after Y years P = starting principal APR = annual percentage rate 1as a decimal2 n = number of compounding periods per year Y = number of years Note that Y is not necessarily an integer; for example, a calculation for six months would have Y = 0.5. Time Out to Think Confirm that substituting n = 1 into the formula for interest paid n times per year gives you the formula for interest paid once a year. Explain why this should be true. example 5 Monthly compounding at 3% You deposit $5000 in a bank account that pays an APR of 3% and compounds interest monthly. How much money will you have after 5 years? Compare this amount to the amount you d have if interest were paid only once each year. Solution The starting principal is P = $5000 and the interest rate is APR = 0.03. Monthly compounding means that interest is paid n = 12 times a year, and we are considering a period of Y = 5 years. We put these values into the compound interest formula to find the accumulated balance, A. A = P * a1 + APR 1nY2 n b = $5000 * a1 + 0.03 112 * 52 12 b = $5000 * 11.00252 60 = $5808.08 For interest paid only once each year, we find the balance after 5 years by using the formula for compound interest paid once a year: A = P * 11 + APR2 Y = $5000 * 11 + 0.032 5 = $5000 * 11.032 5 = $5796.37 After 5 years, monthly compounding gives you a balance of $5808.08 while annual compounding gives you a balance of $5796.37. That is, monthly compounding earns $5808.08 - $5796.37 = $11.71 more, even though the APR is the same in both cases. Now try exercises 63 70. annual Percentage yield (apy) We ve seen that in one year, money grows by more than the APR when interest is compounded more than once a year. For example, we found that with quarterly compounding and an 8% APR, a $1000 principal increases to $1082.43 in one year. This represents a relative increase of 8.24%: relative increase = absolute increase starting principal = $82.43 $1000 = 0.08243 = 8.243% This relative increase over one year is called the annual percentage yield (APY). Note that it depends only on the annual interest rate (APR) and the number of compounding periods, not on the starting principal.